The typical inverse problem of aerodynamics is posed by specifying the pressure distribution on the solid-wall of aerodynamic body and determining the geometry of this solid-wall that realizes this pressure distribution. Generally, the numerical methods for this class of problems are implemented on the Eulerian plane, such as the adjoint method[1], in which we have to firstly estimate the unspecified body shape, then to generate a computing grid around this shape, and to solve the fluid flow governing equations and find the pressure distribution on the body. Next, a very important and time-consuming step is to solve the adjoint equation to modify the geometry shape. This process has to be repeated for several successive steps for shape modification until the final target geometry is reached.
The solving of the flow governing equations, like most current numerical methods did, are implemented on the Cartesian coordinator (the Eulerian plane), where need to generate the computing grid in advance according to the body geometries. For this inverse problem, it needs to re-generate the computing grid since the body geometry keeps being modified. In addition, the smoothness and orthogonality operation for the computing grid, the current numerical methods for this class of inverse problems are very time-consumed.
For inviscid flows, the solid-wall boundary of aerodynamic body is a streamline and any streamline in flow field has its own stream-function with a constant value. It is more suitable to work the unspecified geometry problem on a stream-function plane. There no need to modify the computing grid like that in the adjoint method, since the unspecified shape (solid-wall boundaries) are always represented by streamlines, so that the computing grid will keep on the same in anytime, no matter the geometry of the body shape how to change. This property of solving the geometry shape design problem in the stream-function plane comes to the most efficient process.
However, it was limited in supersonic flows simulation before[2]. When the Euler equations are numerically solved in the stream-function plane, they spatially evolve downstream, there is no any upstream information need to consider, which therefore perfectly follows the physical characteristics in supersonic flows. Until recently, for the subsonic or low speed flow domain, using the advantage of the stream function as one coordinate working on the inverse shape design problems is only for potential flows (inviscid, irrotational, incompressible) and linearized compressible flows[3].
There is an apparent need to the optimal process in shape design. Some industrial applications, such as two-dimensional inviscid subsonic flow simulations and design cases for nozzle and airfoil, are common, useful and necessary in the preliminary phase of product design.